Numerical Simulation of the Ion Etching Process

Numerical techniques traditionally used in the simulation of compressible fluid dynamics are applied to the ion etching process. This process is governed by a non-linear hyperbolic conservation law describing the evolution of the local slope of a surface. The hyperbolic nature of the equation allows discontinuities of slope to develop which are seen numerically and experimentally as cusps in the surface of the etched material. These discontinuities are analogous to shocks in fluid dynamics. Initially, an essentially non-oscillatory (ENO) algorithm is used to simulate the evolution of a single homogeneous material with fixed boundaries and known flux function. The algorithm is then extended to simulate the evolution of two different homogeneous materials which is more representative of a typical etching configuration. The two materials are assumed to be separated by an interface of known form. The additional mathematical and physical reasoning to describe the two-material configuration is presented from which a new algorithm is developed. This algorithm requires the hyperbolic conservation law to be solved on a moving mesh since the interface between the materials is numerically treated as a moving boundary. The nature of the two-material problem is such that shocks and expansion waves can develop at this interface and thus special numerical treatment of the moving boundary is required; this is achieved by using a lower order approximation in this localised region. Finally, a more realistic method to calculate the flux function is adopted which changes the nature of the governing equation since the flux function becomes dependent on the geometry of the surface as well as the local slope. The algorithm is extended to include this flux calculation which allows the numerical simulation of the physically observed phenomena such as RIE lag and undercutting.